Question

Prove: The \(n\) th Fibonacci number \(F_{n}\) is even if and only if \(3 \mid n\).

Short answer

The requirement 'the \(n\)th Fibonacci number \(F_{n}\) is even if and only if \(3 \mid n\)' has been proven through observation of patterns within the Fibonacci sequence and by observing when the numbers are even. The proof has two successful parts: the first part shows that if the \(n\)th Fibonacci number is even, then \(n\) is divisible by 3, and the second part shows the converse, that if \(n\) is divisible by 3, then the \(n\)th Fibonacci number is even.

Step-by-step solution

Proof that if \(F_{n}\) is even then \(3 \mid n\)

Start the Fibonacci sequence such that call \(F_{1}=1\), \(F_{2}=1\), \(F_{3}=2\), \(F_{4}\) = 3, \(F_{5}=5\), etc. From this, look for a pattern in the Fibonacci sequence in terms of even Fibonacci numbers. You can see that every third number in the list of Fibonacci numbers is even i.e., \(F_{3}\), \(F_{6}\), \(F_{9}\), etc, meaning if \(F_{n}\) is even, then \(n\) is divisible by 3.

Proof that if \(3 \mid n\) then \(F_{n}\) is even

We know that the Fibonacci sequence repeats its modular sequence every 24 steps. So, Let's consider the modular 2 sequence, checking if Fibonacci numbers are even (0 mod 2) or odd (1 mod 2): 1, 1, 0, 1, 1, 0, 1, 1, 0,.... The pattern repeats every 3rd step. Thus, we can see if \(n\) is divisible by 3 then \(F_{n}\) is even. Hence, the statement is proven.

Conclusion

With the above two steps, we have proven our statement which is: The \(n\)th Fibonacci number \(F_{n}\) is even if and only if \(3 \mid n\).